Michael Fabricant, the Tory vice-chairman who helped run the Eastleigh campaign, said the party must “focus on the economy, immigration, crime, Europe” and not be distracted by “side issues”.

Mr Fabricant posted a series of messages on the Twitter website, saying the Conservatives needed to do more to “connect” with voters.

“The Conservative voice is muffled and not crisp. It does not clearly project Conservative core policies or principles,” he said. “With Ukip clearly announcing policies the public want to hear, we must do the same.”

A poll showed that 55 per cent of the Eastleigh voters who backed Ukip this week named immigration as the most important issue in the election. Mr Cameron and his allies attempted to play down the Ukip surge as a protest vote with no wider political significance.

But another Cabinet minister described the Eastleigh result as “a seismic moment” that Conservatives must take “very seriously”.

A third minister, a close ally of Mr Cameron, admitted that policies such as gay marriage had contributed to the Eastleigh setback. “We get that has been a problem and we cannot and will not be raising issues like that again which will alienate the grassroots,” he said.

One senior Tory backbencher said: “We’ve insulted Ukip and been soft on the Lib Dems. That has to change and Lynton is the person to do that.”

Mr Crosby, who oversaw the Tory campaign in 2005, is now expected to conduct the party’s Eastleigh “post mortem” and consider how Conservatives can make gains in other Lib Dem seats in 2015.